1/2x+3=6x-8

Solution for 1/2x+3=6x-8 equation:

1/2x+3=6x-8

We move all terms to the left:

1/2x+3-(6x-8)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-6x+8+3=0

We multiply all the terms by the denominator


-6x*2x+8*2x+3*2x+1=0

Wy multiply elements

-12x^2+16x+6x+1=0

We add all the numbers together, and all the variables

-12x^2+22x+1=0

a = -12; b = 22; c = +1;
Δ = b2-4ac
Δ = 222-4·(-12)·1
Δ = 532
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{532}=\sqrt{4*133}=\sqrt{4}*\sqrt{133}=2\sqrt{133}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(22)-2\sqrt{133}}{2*-12}=\frac{-22-2\sqrt{133}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(22)+2\sqrt{133}}{2*-12}=\frac{-22+2\sqrt{133}}{-24}
$